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Robust and efficient implementation of the Delaunay tree

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(1)Robust and efficient implementation of the Delaunay tree Olivier Devillers. To cite this version: Olivier Devillers. Robust and efficient implementation of the Delaunay tree. [Research Report] RR1619, INRIA. 1992, pp.11. �inria-00074942�. HAL Id: inria-00074942 https://hal.inria.fr/inria-00074942 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(295) 9j 4Rg %^k\{'k†k fUb˜gz{„xX

(296) k Uf;k 1q 9f i f 'k x@qZoìk.xXgzfUZk  b wqìxXk 9b0{ fUgzoìkl„c'j'fQf 'gi x tqdxX”k tqÆ{ cLkk¹qxXig  I“ktf@k 9f@k elLqÆ0{ qdxf 'kxUg f@k gyxq iUk¹q C Uk¹xXkt{“fQf N

(297) k Ukgix«{'bì{'ktk  ›b vqìxXk 9bd0{ !g {„xU

(298) k Uf@g b{}| &s'gyxq UbŒq TxXb 9k¹x^f '% k PSxX*f tqxXk*,b 4“ktrkt{'

(299) k Ãq A Vl +'ktL{  bj |„bg {'fÃx^q U)k 9Cb 9g 7 9 j yq t| &s'Ik tqdxXk,b vf UktIk 9b gz{'k¹q „„bg {NfSxg {'f UXb “0j 9k¹xb{  xXboìuk „qUf@Ng 9/j yq )tqxXk¹x¨l k¹qdx !f@gb „qZ0{ C kN| `Œ'kt{f Uk†‘k ebg {“fSxq @)k “ktfUk 9f@k fUb\cekk 1q 9f i 9b gz{'k¹q †lNf 'kt{ f 'k c„q i P9ig =9jNo

(300) x A@g c'gz{'rf 'k\f @gyqZ{'r zMk ›b @oìk úc ˜f 'k¹xXkf @ktuk „bg {'fÃxgixbd{'k\b «f 'uk „q \  yqZ{'+ k “

(301) k zg og f@k ;Cc *f 'sk gz{'kxXj ebUf@g {'rf 'k«f @ktsk ebg {“fSx¨|FnB{ Bq 9ftlf 'k«f@k¹xXwf Ub „bŒxXk  qÆce,b mkwfUbìf@k¹xXfQRg 2”q ebdgz{“f !gix^gz{„xXg “kb Qbj'fÃxXNg “?k Md%eb 71x ^

(302) k  žk Vkt{5Rg   gyxxX0j Aq “ktrdkt{'

(303) k ÃqZfUkf @gyqZ{'r zk| . . .  . O.     

(304) . 'gyxaq zrdbUgzf'o „qdxJcektkt{4k?>ek9f@g=mk

(305)  ”9bC“k*qZ{0ƒtqZ{4c„k+9bom„q@kZBqcmbÃqÆc i %Qgzf f'kwcLk¹xXfV{'b{;gzom zktoìkt{'fÃqZfUgzbd{„x¨|1S„j@f'k

(306) @oìbUklf'gyx$q rb@g f'ok„qdx2f'kwq3ŒqÆ{ŒfSqZrdk b,Qc„ktgz{'› r C V{„qÆogN d›gz{„xXk

(307) @fUgzbd{„xqÆ{0 “k

(308) ktf@g b{„x e:+'gNT gix{'bfsf'kutqxXk.b+“g=Vg“k

(309) qZ{0 9bd{„hmj'

(310) k 5b x kt

(311) k 'gz{'r b{'k¹x¨| &0b6r=g mk qZ{(g “k¹q,b *f 'k „k

(312)  ›b @o

(313) qZ0 { 9k¹x¨lf 'ku$b Ub{'bg “gyqZr ÃqÆo b 2 S --<- SqZ0 { “bdo xXgzfUk¹x.g {qúx@hmj„q Uuk „q kcektkt{ 9bd€o 'j'fUk :g {  -úxXk 9b0{ 'x qÆ0{ Œ@ktoì<b k  d¡g { q ÃqÆ0{ “bo b T“

(314) k  “\g >e

(315) k Ukt{“f  @bdo f 'kúb =“

(316) k ,b ;g {„xX

(317) k @fUgzbd{ e

(318) g { X&s. X.

(319) S0g rjUk . Sgzrdj@k. Sgzrdj@k S. )4 k

(320) iqÆj'{„q ìf@gyqZ{'rdj iqÆf@g b{b 5 --)ÃqÆ{0“bo. xXgzfUk¹x«bd{ qm9gi=A zk. : 2) u$bUb{'bg;“giqÆrSqZo b, S -<--ìxXgzfUk‰x^b{ 5 - -mÃqÆ{“bdo 9 gi=A zk¹x. )4 k

(321) iqÆj'{„q ìf@gyqZ{'rdj iqÆf@g b{ b, 5 --<-)SqZ{0“bdo. xUg f@k¹xg { qìx@hjLq@k.

(322) Sgzrdj@k]X. )2u2bUb{'bg“giqZrÃqÆo. b.  -<-)SqZ{0“bdo. . xXg f@k¹x«g { qg“ktrdkt{'kDSqZfUk‘¨bd{3P„rjSqZfUgzbd{. 

(323) .  .   !#"$% &'()**!#"$+((,#(. -  1()2! -  ./0 $% . Sgzrdj@k. )6.mfSqZfUgixXfUgtxb{ 28-<-<-;xUg f@k¹x^gz{œ“gR>ek

(324) @kt{“f|„bŒxXg f@g b{„x. $.

(325) xXk9b{0'x¨lJf'gyxwfUgzoìk\q@k  b*q .Ž : S |u&P'gix&xXb f7QqUk.gix&qÆqZg iqZc k.gz{ôf'k5/ 'w . gz c Sq  T" %$<&,4B| Sgzrdj @k¹x  f@b Xìcx ',b  xXboìk^ wk

(326) yqZj'{„q ;f@gyqZ{'rdj iqÆf@g b{„x$bQu$bUb{'bdg“giqZrÃqÆo\x$b, xXktf b,webdgz{VfÃx¨%| .mboìk*b 2f 'kto q@kDBqZg  “ktrdkt{Nk

(327) SqZfUk.qÆ{qUkƒ„qÆ{0C zkLQgzf'bj'f*@bdc zkto c bdj vxXb ›f Qq Uk| &s'u k Ubr ÃqÆo „qxcLktkt{j„xXk  b)'qZfSqxXktfSx4xUg {'gi yqsfUb˜f'bŒxXkb,%SgzrjUk¹x  f@b X1l c'j'f,f@gb 9bd€o „qUkwf 'Žk Uk¹xXj zfSx«q i 8f'kxXktfÃx}„qZkxXgtkY2c--<-“|4&s'kxXgzfUk¹xq@k:P0SxUf,gz{„xXk

(328) Uf@k g { )q SqZ0{ “bdo b =“

(329) k †lNqÆ0{ cq ›f@

(330) k q T'x2f '

(331) k !q Uk*q i “

(332) k ktf@k 5gz{qZ{'bdf '

(333) k |SqZ0{ “bo b T“

(334) k t| Sgzrdj @k @k¹xXkt{“fS(x Ãq 9fUg tq 2Uk¹xUj fà x )f 'kxUg tkv,b žf '+k 

(335) k iqÆj'{„q ”&(Uktk,gz{;ceb „ zg {'klf 'k xXg †k.,b  f ')k w

(336) k yqZj'{„q \f @gyqZ{'rdj iqÆf@g b{ g œ { 'qcx 'k  gz{'kl8qÆ0{ gz{ f 'gz› { gz{'kl8qìoìk‰qdxXj @k.b  f'^k 9bmo / k Vg Yf ;,b }f 'kwb „k

(337) ÃqÆf@g b{}| S„b Fgz{„xX

(338) k @fUgzbd{„x¨lVg f^gyx2f 'k{'j'o;c'

(339) k  ,b  VgyxUg f@k ;{'/b “k¹x¨l ›b |“

(340) k ktf@g b{„x$gzf gyx$f 'kw{“j'o.c'

(341) k ,b j'{ 'bŒb Vk 4f UgiqÆ{'r k¹' x  j„x$f 'k{Nj'o.c'

(342) k Sx$b 8f UgiqÆ{'r zk¹x AUk¹qZfUk o“ j Ugz{'r!f 'gizx “

(343) k zktfUgzbdœ {  zj„xf 'mk 9bŒxXf&,b «xXb Uf@g {'r!f ')k ebdgz{VfÃxfUb!ce)k @ktg {„xX

(344) k @f@k e| nBfvgixg €o eb@fSqZ{VfQf@b!{'bdf@Ng 9k*f „qÆ{ f 'k¹xX”k Uk¹xXj zfS x “bŒk¹x{'bŽf “

(345) k ekt0{  xXgzrd{'\g P‚tqÆ{“f i \b{ f 'k “Rg >8

(346) k Ukt{'| f 'qÆfÃq“| J  . O

(347) . 'k5Vgz{06b,vq@bmqA 1kk

(348) b„kg {6f'gix‘„qLk

(349) *›b) wk

(350) yqZj'{„q fUgiqÆ{'rj iqÆf@g b{ tqÆ{ cekj„xXk5 bQoìbŒxXfb,Ff'k%›j'{0'qZoìkt{'fÃq 8xXf@j09f@jUk¹x«g {œ9bom'j'fÃqÆf@g b{„q 8rktbdoktf V|4&P'k g o€eb@fSqZ{VfQf'g {'rŒxq@kf„qZfq  f'ks{'j'oìk

(351) @gNtq 9bdo€'j'fÃqÆf@g b{„xQqUk*c„qxXk5b{ f'ksb@g r< g {„q +“?k P„{'g f@g b{b wf 'k!rktbdoktf @Ng 5bd3c QDk 9fSxtl$f 'Ik 9bd€o 'j'fÃqÆf@g b{q Uk “bd{'kj„xXg {'rúk 1q 9f q @g f 'oìktf@g .qÆ0 { ˜f 'mk “ktrkt{'

(352) k ÃqÆf@mk tqdxUk¹x&q @k.f Uk¹qZfUk úq „qUf†|nBf&gyx{Nk 9k¹x@x@q  fUbqZfUfÃq   k¹q Ax@bd/c kto xX

(353) k „qSqZfU

(354) k i !f@gb “k¹x A@g cLkf '„k 'qZfÃq4gz{qZ{ q @b @gyqZf@ko\qÆ{'{'

(355) k †| &s'kxXfÃqÆ0 { 'q = Lk

(356) Uf@j Uc„qZf@g b{oktf 'Cb '‘x @k¹xXkt{'fsf 'k “gix@q 3qZ{“fSqZrdk,b ŽCÃqdxXf@Ng tq  i g 0{ A@k¹qdxUg {'rúf 'Ik @k 9gyxXgzbd{ {'ktk “k f@b6xXf@b @k!f 'k 'qZfSq1| &P'k Qq ô,b :›b Uo\q zg tgz{'rúf 'k oìktf 'b x“k¹=x AUgzcžk 5qZce,b %k «bdj 

(357) c„kf '‘k “8k Œ

(358) k zb 'okt{mf^,b 2)q “gy=x AUktf@krktbdoktf  5+'Ng T „qÆ0 { C zk¹xf '€k “Ng A@ktfUk.{„qZfU/j Uk,b ,f 'gk 9bmo 'j'f@

(359) k †„| .wxf '€k Ub „k

(360) @fUgzk¹xb «f 'gyxrktboìktf  Qbj N.{'bdf j„xU+ k @k¹q Lq Ugzf 'oktf@Ng Qc'j'f,„q “gix A@ktfUkvbd{'klmf '

(361) k Qbj N4cLŽk Uk¹q  i .k $ zbdgzfSqZc k g {f 'k.q rb @g f 'o

(362) xt| . &P.  D    TIs&N4 TIs&$X4. '| 4|I,bdgix@xXb{'{„qÆfqZ{0 "|4&0ktgi  iqÆj0e| @v{f'k SqZ{0“bdogtkˆ9bd{„xUf@j09fUgzbd{ b.f'k k

(363) iqÆj'{„q fUktk|

(364) „[SRmP@["$  @W$RZY 4

(365) $X[tP  [  ¨[ | &Fbq„k¹qt| .:qÆgi yqZc zkqx|&0k'{'gNtq J`Qk

(366) „bUfvnY`vnY. 2<2 : -“| '| 4N | I,bgyx@xXb{'{„qÆfqZ0{  "s| &0ktg i yqZ0j eŒ | . 'gzk

(367) Sq=T'gNtq +@k

(368) Uk¹xUkt{'fSqZfUgzbd{ b, b0c QDk 9fSx ) f ')k w

(369) k yqZj'{„q I&(@ktk|$nB { J[ SR 

(370) ¹ Y 4„R & 

(371) Y R  $RZY 4

(372) "! $µW $ ¥R NW $#w[SRZY5"[ $p%P A '&LRm)P ( $µ"R *@,+4 [ ¶T $7&9l dj'{'Mk 28&<$X1| . &.

(373) D" &N4. @*|X kVgi  zk

(374) Sx¨l0.„| "ktgixXk

(375) tl1qÆ{0 "|X&Fktg i yqZj0e|S„j  i )C V{„qÆogN* k

(376) iqZj'{„q *fUg9 qZ{'rdj iqÆf@g b{sgz{„ brmqUgzf'ogNak$„k9f@k&f@g ok'„k

(377) }bek

(378) ÃqÆf@g b{}| RZY 4

(379) µW ¥R NW #&[ RZY\[ µP

(380) N[SRmP%\W   4 4   W ¥R 9|+&Fbq„k‰q†|.:Œ qZg iqÆc zkqx &0k'{'g9 tq e`Qk

(381) Lb@f,nKw`vnK. 2  : “ & |X.vc„xXfSq9f''jNc gixc'k;gz{ / %6H. S 28&5d š ` .Ž .  &Fd2 l qZj'rdj„xXf*2c&&F2Se |. "$. . % $   &. $ $ . nS|<'aogiUgixaqÆ{0 '|$6,qZ{'{ |C.rkt{'k

(382) Sq Nq@bmqA4fUb^@ktoìb<Œgz{'rz“ktrkt{'k

(383) Sq9gzk¹x¨| 2c&&F2d6 | "˜qÆ{“j„x A@g 'f†l v{'=g 

(384) k ÃxXgzYf ,b '6^q zRg ›b U{'giq“al I,

(385) k 

(386) k z

(387) k | T')*" +,&-4›32f| '4„xXfUk†g {}l .*| )&{'gzr “ftl '| " q !l &&!| vrdj Vkt{}l8qZ0{  p`s| .Vq ?ž |  *2RmP !  [   tRmP ¨RZ Y 4

(388) $µW $ ¥R 'W 2T1[ RZY\"[ $µP ¹| &0k '{'Ng tq &`

(389) k „bUf .6H. Z&H` 72c$-1l .$A'bb 8b }9bm o 'jNfU

(390) k vx 9gzkt0{ 9k‚l 6,q  zktfUb{ v{'9g Œ

(391) k ÃxXgzYf l .Œ

(392) k 'fUkto.cL

(393) k *28&<&-1| T'" &<-,4 1 | '}“

(394) k yxXc @j'{'{'

(395) k .qZ0{  'Q| 3a| " 0j , [mk| .mgzoìj iqÆf@g b{ôb xUg €o  g 9g f )qf@k A0 {'gihmj'kìf@œ b 9b „kMQgzf  “ktrkt{'

(396) k SqZfUIk tqxXk¹xg {6rktbdoktf @Ng q zrdb Ugzf 'o

(397) x¨|  . LPXW  &†W $ ¥R  &R  #PXW 4  "&9F l & ) XX 2c- : l mqZ{Vj„q  28&<&-1| TSLb  $&4 .„| ';| S'b @fUj'{'k | .mfÃqÆc zk*o

(398) qZg {'f@kt{„qÆ{ 9kb 4ebg {“fwxXktf&f UgiqÆ{'rj yqZfUgzb{„xg { 7f ^b “gzoìkt{„xUg b{„x¨|nB{

(399) ¹ Y 41R & 

(400) mY R  R

(401)  ŒW $ ¥R  &5R  RZ Y 4

(402) $X[tP 

(403)  ! [  †[ Xl „qZrk¹x : & :  : &&1;l 28&<$&1| TSS<q $&4 '6 | S0SqZ{9 b{} | &0b 'b zbdrg k “gy=x Akt fUk\ktf\q rb Ugzf 'ogyhmj'kr Ãq 'gihmj'k| ¶TVP@[ ! # zR 

(404) "i[ l XF2 ) &32^ &<X1; l .*@g N28&<$&1| dbdj @+{ kt1 k¹ x .%SW66'}&  U&W` @Ž3/.Ž “)k ok 1  9kto c UMk 28&<$$„q  &0bj zbdj„xXk| (U].0.^$<&,4 /2| ' | Uj'gzc„qdx¨4l .| .Vq k¹xUg {}l$qZ0{  '| .mfUb RP | '4„xXg zbd{ rdktboìktf  ).c'j'ig “g {'r @bdc'j„xXf;q rb @g f 'o

(405) D x  @bdo g €o Uk 9gixXIk 9bmo 'j'fÃqÆf@g b{„x¨|˜nB{ $   ¹Y ! 4LR & 

(406) mY R  $RZY 4

(407) $µW $¥R NW  #&[ RZY\"[ $µP    8WmWP  ?P ,

(408)  (Œ[ “/l LqÆrk¹x 5 -<$ 5 2 Œl dj'{'M k 28&<$&1| (UWVD$X4 . | *%| U Uktkt{'kôqÆ0{  S,N| S,N| V«qÆbN| S0g {'gzfUk G@k¹xXb j'f@g b{ 9bmo 'j'fÃqZfUgzbd{„q rk  boìktf  V|

(409) np{ 

(410) ¹ Y 4„R & 

(411) Y R  aR

(412) ŒW $¥R  &R $RÆ Y 4

(413) $D[tP  › [  ¨[ l „qZrdk¹Dx 2 :  f 2 S 5 l 28&<$X1| T" g C$&4 u.| " g zkt0{ Œb Œg Z)| bdj'/c €k Uk 9gixXg b{6rdktboìktf  )*q!rdkt{'

(414) k Ãq fUk A'{'gihjNk  b  tq N9 j iqÆf@g {'Ir zg {'kìqZ0{ 6xXktrdokt{“fvg {'f@

(415) k ÃxXk 9fUgzbd{„xj„xUg {'Ir Ubj'0{ “k  q Ugzf 'oktfUg Z| nB{

(416) † Y 4'R & 

(417) mY R  FR

(418) mW $ ¥R  &^R $RZY 4

(419) $X[tP   [  †[ dl „qÆrk¹x S -<- S - S ;l 28&<$&1| T"œW$<&,4 )ì,| "

(420) k  i'b @{;qÆ0{ !.„| ƒq , '

(421) k t| "!}W    PSWmP vRv[$# › [  $ ŒW $µW $  4e[ &^W  W  TVR<P  $ V= Y &9X| &Jk ='{'g tq d`Q

(422) k ebUf &` . - :  $<&“l {'9g 

(423) k ÃxXgzdf qZ, !f “k¹x .Vqq  yqZ0{ “k¹x¨l .Vqq @c 0j , kt{}al 28&<$&1| T's66&F24. 28-.

(424) N| vgzk Œk Urk f†l a| 'b @{}l"|/“kY/8b@kt{tg¡l6v|/.voìo\qZ{'{}lLqÆ{ .*|fI'j', {'r rk t| ô_ 8R<$ 2WmP [ †RmPJT“[SRZY5["$pP< tRÆY 4

(425) $pW $ ¥R V|&0kd'{'gNtq Œ`Qk

(426) „bUf 28X  l nB{„xUfUgzfUj'f j 'ktb Ugix 'knB{3›bUo\qÆf@g=žl'}& *l„jUgT}l dj  2c&&F2| "o | &Fktg i yqZ0j e| e[tP & “['& W TŒRmP<%$ VY5[ & PSW  ŒRZY  &[' &   NWZY  

(427) '['& [  T [t RZ. Y "[  $µ"P ›[*W  TVRm"P  $ VY 

(428) 

(429) '[Ãs| wk 9ktoìc'

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(431) k ÃxXgzf ^k 1 3aq Ugix  .m0j e;l d S0SqZ0{ 9k eÃ|. *.$0/1+,&324   D

(432) /3 .$A 

(433)   & " * D, |&P  A. Fktg_&F2 4 &. 22.

(434)

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